Theorem fusgrvtxfi 29909

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29908	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 503	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  Fincfn 8981  Vtxcvtx 29583  USGraphcusgr 29739  FinUSGraphcfusgr 29906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-fusgr 29907

This theorem is used by:  fusgrfupgrfs  29921  nbfusgrlevtxm1  29967  nbfusgrlevtxm2  29968  nbusgrvtxm1  29969  uvtxnm1nbgr  29994  cusgrm1rusgr  30172  wlksnfi  30505  fusgrhashclwwlkn  30679  clwwlkndivn  30680  fusgreghash2wsp  30948  numclwwlk3lem2  30994  numclwwlk4  30996